In my memory, the original X-bomb (where X was a letter representing a word too offensive to be repeated in full) was certainly f-bomb. I would have guessed that the term was at least 20 to 25 years old, and that it originated in newspaper, radio, an TV accounts of something a celebrity, politician, or athlete had said, in the course of which the quoted person had intentionally said or inadvertently let slip the word fuck.
The journalistic milieu for X-bombs
In those days, mainstream print journalism avoided including certain obscenities and profanities in quotations even though the person quoted had used them. Their reason was chiefly to avoid needlessly antagonizing readers who might find the crudeness of the wording offensive. The entry for "obscenities, profanities, vulgarities" remained unchanged in editions of The Associated Press Stylebook and Libel Manual published between 1980 and 1994 (I have the editions of 1980, 1987, and 1994). The successor stylebook, The Associated Press Stylebook and Briefing on Media Law (2002) alters the wording only slightly and in very few places. Here is the wording from the 1987 edition of the AP Stylebook:
obscenities, profanities, vulgarities Do not use them in stories unless they are part of direct quotations and there is a compelling reason for them.
When a profanity, obscenity or vulgarity is used, flag the story at the top:
Editors: Language in the 4th graf may be offensive to some readers.
Then confine the offending language in quotation marks, in a separate paragraph that can be deleted easily by editors who do not want it.
In reporting profanity that normally would use the words damn or god, lower-case god and use the following forms: damn, damn it, goddamn it. Do not, however, change the offending words to euphemisms. Do not, for example, change damn it to darn it.
If a full quote that contains profanity, obscenity or vulgarity cannot be dropped but there is no compelling reason for the offensive language, replace letters of an offensive word with a hyphen. The word damn, for example, would become d--- or ----.
When the subject matter of a story may be considered offensive, but the story does not contain quoted profanity, obscenity or vulgarities, flag the story at the top:
Editors: The contents may be offensive to some readers.
The handling of full quotes containing vulgar language changed by 2004, however. In the entry for "obscenities" in The UPI Stylebook and Guide to Newswriting, fourth edition (2004)—which repeats many AP Stylebook entries verbatim—has this paragraph at the close of the entry:
If a full quote that contains offensive language cannot be dropped but there is no compelling reason for the offensive language, replace it with expletive in parentheses: "My opponent is a (expletive) liar," the candidate said. Do not use the term expletive deleted.
In a journalistic landscape where the profession's primary stylebook recommended replacing fuck with f--- on the rare occasions when the term had to be included in a quotation, it's easy to see how f-bomb as a term might have caught on among journalists. But for much of the period between 1980 and 1994 (and perhaps after), AP seems not to have n----- a suitable euphemism for nigger.
In fact, the entry for "nationalities and races" in the AP Stylebooks during the same period is rather startlingly off-hand by comparison to its treatment of "obscenities, profanities, vulgarities":
nationalities and races Capitalize the proper names of nationalities, peoples, races, tribes, etc.: Arab, Arabic, African, American, Caucasion, Cherokee, Chinese (both singular and plural), Eskimo (plural Eskimos), French Canadians, Gypsy (Gypsies), Japanese (singular and plural), Jew, Jewish, Latin, Negro (Negroes), Nordic, Sioux, Swede, etc.
Lowercase black (noun or adjective), white, red, mulatto, etc. See colored [where the entry says, "In some societies, including the United States, the word is considered derogatory and should not be used"].
See race for guidelines on when racial identification is pertinent in a story.
Lowercase derogatory terms such as honky and nigger. Use them only in direct quotes when essential to the story [to which the 2002 edition adds "and flag the contents in an editor's note"].
I call this treatment "rather startlingly offhand" because I think that Matt Mitchell,"The Fine Art of Defusing an N-Bomb," in Can I Teach That?: Negotiating Taboo Language and Controversial Topics in the Language Arts Classroom (2016), accurately assesses the differing taboos associated with the two main X-bombs today:
While few high school students will be scandalized by encountering "adult language" (or "adult content") in a literary text assigned for classroom study (in fact, for many of them, it will render the text appealingly "real"), many will—for good reason—be uncomfortable with the appearance of racial slurs in those same texts. The N-bomb carries a much more inflammatory and potentially disruptive charge these days than the F-bomb.
During this same period—and continuing to the present today—in the United States, radio and television broadcasts were subject to penalties for using certain unacceptable words on the air. As the U.S. Federal Communication Commission's Obscene, Indecent and Profane Broadcasts page observes,
Federal law prohibits obscene, indecent and profane content from being broadcast on the radio or TV.
The key word here is broadcast—as in "sent over the public airways." No such prohibition applies to cable TV transmissions or Internet podcasts or streaming Internet video and audio. The distinction has always had a powerful odor of arbitrariness and irrationality to it, and it has not aged well as first computers and then smartphones proliferated and brought Internet content of all sorts within reach of most people from a very young age.
But the specter of legal repercussions from allowing a proscribed word to appear in a broadcast helped turn fear of a slip-up into a kind of fetish in broadcast journalism, especially as those words entered the everyday vocabularies of more and more people. It is in this milieu, I believe, that terms of the type X-bomb caught on.
Early sightings in the wild
The earliest reported mention of f-bomb I've been able to find is from 1988, as cited in Jesse Sheidlower, The F-Word (2009), which offers an extensive and useful compilation of instances in its entry for the term:
F-bomb noun the word FUCK or one of its variants or compounds, esp. with reference to it as a shocking or inappropriate term. Often in to drop the F-bomb. [First three cited occurrences:] 1988 Newsday (New York) (Aug. 11) i 144/3: That was when I used to use the F-bomb. 1991 Boston Globe (Jan. 11) (Sports section) 31: It looked like I was calling him everything in the book.... I dropped the F-bomb a few times. 1995 Denver Post (June 11) B 10/3: I tried not to say any F-bombs with my mom there.
Meanwhile the earliest instance of n-bomb in the relevant sense is the one from 1995 that JOSH notes in his answer. From the entry for "Mark Twain" in An Encyclopedia of Swearing: The Social History of Oaths, Profanity, Foul Language, and Ethnic Slurs in the English-Speaking World (2006):
Huckleberry Finn remains a controversial text depicting the struggles of innocence to "do the right thing" in a corrupted and unjust society, especially because its direct language, notably the use of nigger, has become increasingly provocative and unacceptable. On the latter point the noted theater critic Frank Rich referred in 1995 to "Dropping the N-Bomb."
Rich's article of that title appeared in the New York Times on March 16, 1995, page A15. (My thanks to JOSH for pointing out this instance in his answer.) It may be that Rich was simply invoking an existing term (n-bomb) in his title; or he may have been riffing on the already-established term f-bomb, whose usage is documented to 1988, as noted above.
A complicating factor: f-word and n-word
One complication in this whole investigation is that "f-word" and "n-word" very likely entered the language before their immediate counterparts f-bomb and n-bomb did—and these X-word formulations served essentially the same purpose of telegraphing the actual words without saying them that the X-bomb expressions did. From "Calendar: 'Hi Strung'" in the Santa Monica [California] College Corsair](http://cdnc.ucr.edu/cgi-bin/cdnc?a=d&d=CRS19781011.2.13&txq=%22f-word%22) (1978):
TOMORROW "High Strung" will appear in the Amphitheater at 11 a.m. The A.S. Commission will meet in the cafeteria board room at 3 p.m. "Shock," "F-Word," and "The Simpletones" will play at Club 88, 11784 West Pico Blvd. in West Los Angeles at 9 p.m. Must be 21. Admission is $1.50.
Los Angeles in 1978 was a hotbed of early Germs/Black Flag/Circle Jerks–style punk rock, and the band names Shock, F-Word, and Simpletones suggest a kindred aesthetic. Hence, I take the band name F-Word to be an early instance of using f-word to stand for fuck. An Elephind search turns up seven additional instances of f-word from the 1980s prior to 1988, starting in 1983.
And on the n-word side of the ledger, from Edwin Glosson, "What Is Our Role in Making the 'N' Word Acceptable?" in the San Antonio [Texas] Register (December 10, 1992):
When you go to the movies, our favorite actors, are on the silver screen using the "N" word.
All around the world Anglos and other races, are hearing the music and seeing the movies where the "N" word is being used freely.
If we don not respect ourselves, how do we expect others to? We must call on the entertainment industry to change its scripts and the record industry to change the lyrics of their songs when the word "N" is used. ...
When you look in the mirror ask yourself, "when was the last time I used the "N" word?" It may have been sooner than we think. It is time for a change. But it will not happen until we demand it. We must respect ourselves, if we want others to respect us. Let's make the change today.
In the publications I've consulted, the earliest confirmed occurrences of f-word, f-bomb, n-word, and n-bomb are as follows: f-word, 1978; f-bomb, 1988; n-word, 1992; n-bomb, 1994. So in this four-way contest, f-word wins the overall competition for earliest term, and f-bomb wins for earliest X-bomb term.